📂Stochastic Differential Equations

Ito Formula and Martingale Representation Theorem

Theorem ^{1 2}

Given a probability space $( \Omega , \mathcal{F} , P)$ and a filtration $\left\{ \mathcal{F}_{t} \right\}_{t \ge 0}$, let’s say that a Wiener process $\left\{ W_{t} \right\}_{t \ge 0}$ is $\mathcal{F}_{t}$-adapted.

Itô’s Lemma

If $f \in \mathcal{L}^{2} (P)$, then there exists a unique stochastic process $X (t,\omega) \in m^{2}(0,T)$ satisfying: $$f (\omega) = E (f) + \int_{0}^{T} X(s, \omega) d W_{s}$$

Martingale Representation Theorem

For all $t \ge 0$, if $f_{t} \in \mathcal{L}^{2} (P)$ and $f_{t}$ is a $\mathcal{F}_{t}$-martingale with respect to probability $P$, then for all $t \ge 0$, there exists a unique stochastic process $X (t,\omega) \in m^{2}(0,t)$ satisfying: $$f_{t} (\omega) = E \left( f_{0} \right) + \int_{0}^{t} X(s, \omega) d W_{s}$$

Explanation

$\mathcal{L}^{p} (\mu)$ is a Lebesgue space containing functions $f$ that satisfy $\displaystyle \int_{\Omega} \left| f \right| ^{p} d \mu < \infty$ under the Lebesgue measure $\mu$. Since the given probability space $( \Omega , \mathcal{F} , P)$ treats probability $P$ also as a measure, in the above proposition $f \in \mathcal{L}^{2}(P)$ is a $P$ integrable function, and thus the notation of expected value $\displaystyle E(f) = \int_{\Omega} f d P$ akin to $F$ could emerge.

Itô’s lemma is regarding a fixed time $T$, whereas the Martingale Representation Theorem is for all $t \ge 0$. Note that $f_{t}$ being a $\mathcal{F}_{t}$-martingale means $f_{t}$ is $\mathcal{F}_{t}$-adapted and satisfies: $$\forall t \ge s \ge 0, E \left( f_{t} | \mathcal{F}_{s} \right) = f_{s}$$ In this definition, it’s seen that the expected values appearing in the Martingale Representation Theorem need not specifically be $E \left( f_{t} \right)$, $E \left( f_{0} \right)$ is also sufficient.